Let be the ellipse . The line intersects at two distinct points . Let be the -coordinates of . If , find .

Analytic Geometry Solution – Problem 30: find

Question

Let $E$ be the ellipse $\dfrac{x^2}{25}+\dfrac{y^2}{16}=1$ . The line $l: y=mx+1$ intersects $E$ at two distinct points $A,B$ . Let $x_1,x_2$ be the $x$ -coordinates of $A,B$ .

If $x_1x_2=-15$ , find $m$ .

Step-by-step solution

(1) Substitute $y=mx+1$ into $\dfrac{x^2}{25}+\dfrac{y^2}{16}=1$ : $\frac{x^2}{25}+\frac{(mx+1)^2}{16}=1.$ Multiply by $400$ : $(16+25m^2)x^2+50mx-375=0.$ (2) By Vieta's formulas for the roots $x_1,x_2$ , $x_1x_2=\frac{-375}{16+25m^2}.$ Given $x_1x_2=-15$ , we get $\frac{375}{16+25m^2}=15\Rightarrow 16+25m^2=25\Rightarrow m^2=\frac{9}{25}.$ Therefore $m=\pm\frac{3}{5}.$

Final answer

The slope is $m=\pm\dfrac{3}{5}$ .

Marking scheme

Step 1 — Setup

Checkpoint: correctly substitute the line into the ellipse equation and obtain a quadratic in $x$ (2 pts)

Step 2 — Key Calculation

Checkpoint: apply Vieta to express $x_1x_2$ and solve $\frac{-375}{16+25m^2}=-15$ (3 pts)

Step 3 — Final Answer

Checkpoint: report $m=\pm\frac35$ (2 pts)

Zero credit if: uses Vieta with incorrect leading coefficient.

Deductions: -1 pt for algebra simplification error if the quadratic setup is correct.

Analytic Geometry – Problem 30: find