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If the straight line xcosalpha+ysinalpha...

If the straight line `xcosalpha+ysinalpha=p` touches the curve `(x^2)/(a^2)+(y^2)/(b^2)=1` , then prove that `a^2\ cos^2alpha+b^2\ sin^2alpha=p^2` .

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We have,
`x cos alpha+y sin alpha=p ldots . .(i) `
`frac{x^{2}}{a^{2}}-frac{y^{2}}{b^{2}}=1 ldots ldots(i i)`
As, the straight line (i) touches the curve (i i).
So, the straight line (i) is tangent to the curve (i i).
Also, the slope of the straight line, `m=frac{-cos alpha}{sin alpha}`
...
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RD SHARMA-TANGENTS AND NORMALS-Solved Examples And Exercises
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  13. Find the angle of intersection of curve y=4-x^2 and y=x^2

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  14. Show that y=x^3 and 6y=7-x^2 intersect orthogonally:

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  15. Show that x^3-3x y^2=-2 and 3x^2\ y-y^3=2 intersect orthogonally:

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  16. Show that x^2+4y^2=8 and x^2-2y^2=4 intersect orthogonally

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  17. Show that x^2=4y and 4y+x^2=8 intersect orthogonally at (2,\ 1)

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  18. Show that x^2=y and x^3+6y=7 intersect orthogonally at (1,\ 1)

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  19. Show that y^2=8x and 2x^2+y^2=10 at (1,\ 2sqrt(2))

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