A. M(-1, 4) and N(2, -3) are two fixed points. A variable point S(x, y) moves such that 2|MS| = |NS|. Find the equation of the locus of S expressed in its simplest form, and state its constant properties. ​b. Find the points in which the locus meets the y-axis. ​c. Find the equation of the tangent to the parabola 3y2 = 16x at the point (3, 4).

Computer Science

ScanSolve AI Answer

Answers

3 parts
a

3x2+3y2+12x−38y+55=03x^2 + 3y^2 + 12x - 38y + 55 = 0

(−2,193)\left(-2, \frac{19}{3}\right)

2323=2583\frac{\sqrt{232}}{3} = \frac{2\sqrt{58}}{3}

b

(0,11) and (0,53)(0, 11) \text{ and } \left(0, \frac{5}{3}\right)

c

2x−3y+6=02x - 3y + 6 = 0

Step-by-step solution

4 steps

Part a

Find the equation of the locus of S expressed in its simplest form, and state its constant properties.

From our previous conversation, we found:
The equation of the locus of S is 3x2+3y2+12x−38y+55=0\boxed{3x^2 + 3y^2 + 12x - 38y + 55 = 0}.
The constant properties are:
Center: (−2,193)\boxed{\left(-2, \frac{19}{3}\right)}
Radius: 2323=2583\boxed{\frac{\sqrt{232}}{3} = \frac{2\sqrt{58}}{3}}

Part b

(0,11) and (0,53)(0, 11) \text{ and } \left(0, \frac{5}{3}\right)

Find the points in which the locus meets the y-axis.

From our previous conversation, we found:
The points are (0,11) and (0,53)\boxed{(0, 11) \text{ and } \left(0, \frac{5}{3}\right)}.

Part c

2x−3y+6=02x - 3y + 6 = 0

Find the equation of the tangent to the parabola 3y2=16x3y^2 = 16x at the point (3,4)(3, 4).

  1. Verify the point lies on the parabola.

    Substitute (3,4)(3, 4) into 3y2=16x3y^2 = 16x:
    3(4)2=3(16)=483(4)^2 = 3(16) = 48
    16(3)=4816(3) = 48
    Since 48=4848 = 48, the point (3,4)(3, 4) lies on the parabola.

  2. Differentiate the equation of the parabola implicitly with respect to xx.

    3y2=16x3y^2 = 16x ddx(3y2)=ddx(16x)\frac{d}{dx}(3y^2) = \frac{d}{dx}(16x) 6ydydx=166y \frac{dy}{dx} = 16 dydx=166y=83y\frac{dy}{dx} = \frac{16}{6y} = \frac{8}{3y}

  3. Find the slope of the tangent at the point $(3, 4)$.

    Substitute y=4y=4 into the derivative:

    m=dydx∣(3,4)=83(4)=812=23m = \frac{dy}{dx}\Big|_{(3,4)} = \frac{8}{3(4)} = \frac{8}{12} = \frac{2}{3}

  4. Use the point-slope form to find the equation of the tangent.

    The point-slope form is y−y1=m(x−x1)y - y_1 = m(x - x_1).
    Using (x1,y1)=(3,4)(x_1, y_1) = (3, 4) and m=23m = \frac{2}{3}:

    y−4=23(x−3)y - 4 = \frac{2}{3}(x - 3)

    Multiply by 3 to clear the fraction:

    3(y−4)=2(x−3)3(y - 4) = 2(x - 3) 3y−12=2x−63y - 12 = 2x - 6

    Rearrange into the general form Ax+By+C=0Ax + By + C = 0:

    2x−3y+12−6=02x - 3y + 12 - 6 = 0 2x−3y+6=02x - 3y + 6 = 0

    The equation of the tangent is 2x−3y+6=0\boxed{2x - 3y + 6 = 0}.

Was this helpful?

Try one yourself

A new problem that tests the same skill.

Still stuck on this one?

Ask a follow-up, and the answer starts from this question.

Got a different question?Ask your own question

More Computer Science Questions

QM(−1, 4) and N(2, −3) are two fixed points. A variable point S(x, y) moves such that 2|MS| = |NS|…QInstructions: Answer any five (5) Questions. If F(−1,−4) and H(−2,−3) are two fixed points. A…QA circle has its centre at M(−2; 1) and passes through S(1; −2). Determine the coordinates of M and…QGiven points P(2, 9), Q(6, 3), R(2, −3), and S(−4, 3), and the equation y = 1/2x + 3, plot the…QA. Use Lagrange Multiplier (LM) to find points on the surface z=x2+y2 closest to the point (0,0,2)…
See all Computer Science questions →
Got a syllabus? Photograph it and I'll tell you what's due and what's worth the most.
Handwritten step-by-step solution preview

Handwritten Step-by-Step Solution

Get this as a handwritten image on WhatsApp — like a tutor wrote it for you.

Try on WhatsApp